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14 The Dynamical Equations of Cosmology
We take the cosmic fluid to be co-moving as we discussed in Chap. 13. From the
definition of the 4-velocity and the FLRW metric we then obtain the 4-velocity of
the fluid
u
μ
=
dx
μ
ds
= (1, 0, 0, 0), u β = g βα u
α
= (1, 0, 0, 0).
(14.2)
From this the energy-momentum tensor on the right side of the field equations is
T
μ ν = (ρ + p)u
μ u ν − pδ
μ
ν =
⎛
⎜
⎜
⎝
ρ 0 0 0
0 − p 0 0
0 0 − p 0
0 0 0 − p
⎞
⎟
⎟
⎠ .
(14.3)
This is a wonderfully simple form for the right side of the field equations.
To get the geometric left side of the field equations we need the Einstein tensor.
This involves slightly tedious but straight-forward algebra, so we have relegated it to
Appendix 1. and Exercises 14.1 and 14.2. The diagonal components of the Einstein
tensor are the following simple functions of the scale factor a(t) and its derivatives,
G
0
0 = −3
k
a 2 +
a
2
c 2 a 2
, a
≡
da
dt
,
G
1
1 = G
2
2 = G
3
3 = −
k
a 2 +
a
2
a 2 c 2 +
2a
ac 2
,
(14.4)
and the off-diagonal components are identically zero. We substitute these and the
energy-momentum tensor (14.3) into the field equations, to obtain
−
8π G
c 4
ρ = − 3
k
a 2 +
a
2
c 2 a 2
,
(14.5a)
8π G
c 4
p = −
k
a 2 +
a
2
+
2a
ac 2
.
(14.5b)
These Einstein field equations are the basis of standard cosmological theory. They are
widely referred to as the Friedmann equations. A prime task of theoretical cosmology
is to solve them for the scale factor. To do this we need to choose an appropriate
source density and pressure or some relation between the two.
Three prime tasks of observational cosmology are to determine the curvature
parameter k and the value of the cosmological constant in (13.5), and of course
compare observations with theory.
14 The Dynamical Equations of Cosmology
We take the cosmic fluid to be co-moving as we discussed in Chap. 13. From the
definition of the 4-velocity and the FLRW metric we then obtain the 4-velocity of
the fluid
u
μ
=
dx
μ
ds
= (1, 0, 0, 0), u β = g βα u
α
= (1, 0, 0, 0).
(14.2)
From this the energy-momentum tensor on the right side of the field equations is
T
μ ν = (ρ + p)u
μ u ν − pδ
μ
ν =
⎛
⎜
⎜
⎝
ρ 0 0 0
0 − p 0 0
0 0 − p 0
0 0 0 − p
⎞
⎟
⎟
⎠ .
(14.3)
This is a wonderfully simple form for the right side of the field equations.
To get the geometric left side of the field equations we need the Einstein tensor.
This involves slightly tedious but straight-forward algebra, so we have relegated it to
Appendix 1. and Exercises 14.1 and 14.2. The diagonal components of the Einstein
tensor are the following simple functions of the scale factor a(t) and its derivatives,
G
0
0 = −3
k
a 2 +
a
2
c 2 a 2
, a
≡
da
dt
,
G
1
1 = G
2
2 = G
3
3 = −
k
a 2 +
a
2
a 2 c 2 +
2a
ac 2
,
(14.4)
and the off-diagonal components are identically zero. We substitute these and the
energy-momentum tensor (14.3) into the field equations, to obtain
−
8π G
c 4
ρ = − 3
k
a 2 +
a
2
c 2 a 2
,
(14.5a)
8π G
c 4
p = −
k
a 2 +
a
2
+
2a
ac 2
.
(14.5b)
These Einstein field equations are the basis of standard cosmological theory. They are
widely referred to as the Friedmann equations. A prime task of theoretical cosmology
is to solve them for the scale factor. To do this we need to choose an appropriate
source density and pressure or some relation between the two.
Three prime tasks of observational cosmology are to determine the curvature
parameter k and the value of the cosmological constant in (13.5), and of course
compare observations with theory.